# Properties

 Label 8.0.378535936.1 Degree $8$ Signature $[0, 4]$ Discriminant $378535936$ Root discriminant $11.81$ Ramified primes $2, 19$ Class number $1$ Class group trivial Galois group $Q_8:C_2$ (as 8T11)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 20*x^6 - 44*x^5 + 98*x^4 - 128*x^3 + 144*x^2 - 96*x + 34)

gp: K = bnfinit(x^8 - 4*x^7 + 20*x^6 - 44*x^5 + 98*x^4 - 128*x^3 + 144*x^2 - 96*x + 34, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34, -96, 144, -128, 98, -44, 20, -4, 1]);

$$x^{8} - 4 x^{7} + 20 x^{6} - 44 x^{5} + 98 x^{4} - 128 x^{3} + 144 x^{2} - 96 x + 34$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$378535936$$$$\medspace = 2^{20}\cdot 19^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $11.81$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 19$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $4$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{85} a^{6} + \frac{7}{17} a^{5} + \frac{2}{85} a^{4} - \frac{6}{85} a^{3} - \frac{12}{85} a^{2} - \frac{33}{85} a - \frac{1}{5}$, $\frac{1}{85} a^{7} - \frac{33}{85} a^{5} + \frac{9}{85} a^{4} + \frac{28}{85} a^{3} - \frac{38}{85} a^{2} + \frac{33}{85} a$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $3$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-\frac{2}{85} a^{7} - \frac{19}{85} a^{5} - \frac{18}{85} a^{4} - \frac{56}{85} a^{3} - \frac{9}{85} a^{2} - \frac{66}{85} a + 1$$ (order $8$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$55.0761610673$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 55.0761610673 \cdot 1}{8\sqrt{378535936}}\approx 0.551492474014$

## Galois group

$D_4:C_2$ (as 8T11):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 16 The 10 conjugacy class representatives for $Q_8:C_2$ Character table for $Q_8:C_2$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Galois closure: Deg 16 Degree 8 siblings: 8.4.136651472896.2, 8.0.8540717056.3

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

gp: idealfactors = idealprimedec(K, p); \\ get the data

gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

magma: idealfactors := Factorization(p*Integers(K)); // get the data

magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.20.55$x^{8} + 4 x^{6} + 4 x^{5} + 6 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2} 1919.4.2.2x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.76.2t1.a.a$1$ $2^{2} \cdot 19$ $$\Q(\sqrt{19})$$ $C_2$ (as 2T1) $1$ $1$
* 1.8.2t1.b.a$1$ $2^{3}$ $$\Q(\sqrt{-2})$$ $C_2$ (as 2T1) $1$ $-1$
1.152.2t1.b.a$1$ $2^{3} \cdot 19$ $$\Q(\sqrt{-38})$$ $C_2$ (as 2T1) $1$ $-1$
1.19.2t1.a.a$1$ $19$ $$\Q(\sqrt{-19})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.8.2t1.a.a$1$ $2^{3}$ $$\Q(\sqrt{2})$$ $C_2$ (as 2T1) $1$ $1$
* 1.4.2t1.a.a$1$ $2^{2}$ $$\Q(\sqrt{-1})$$ $C_2$ (as 2T1) $1$ $-1$
1.152.2t1.a.a$1$ $2^{3} \cdot 19$ $$\Q(\sqrt{38})$$ $C_2$ (as 2T1) $1$ $1$
* 2.1216.8t11.a.a$2$ $2^{6} \cdot 19$ 8.0.378535936.1 $Q_8:C_2$ (as 8T11) $0$ $0$
* 2.1216.8t11.a.b$2$ $2^{6} \cdot 19$ 8.0.378535936.1 $Q_8:C_2$ (as 8T11) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.